On a class of doubly nonlinear nonlocal evolution equations

Abstract

This note deals with the initial value problem for the abstract nonlinear nonlocal equation $ (\mathcal A u)' + (\mathcal B u) \ni f$, where $ \mathcal A $ is a possibly degenerate maximal monotone operator from the Hilbert space $ V $ to its dual space $ V ^* $, while $ \mathcal B $ is a nonlocal maximal monotone operator from $ L^2(0,T,V) $ to $ L^2(0,T;V^*)$. Assuming suitable boundedness and coerciveness conditions and letting $ \mathcal A $ be a subgradient, existence of a solution is established by making use of an approximation procedure. Applications to various classes of degenerate nonlinear integrodifferential equations are discussed.